The (narrow-band) CAF is generally given as
            CAF      ⁡              (                  τ          ,          δ                )              =                  ∫        0        T            ⁢                                    s            1                    ⁡                      (            t            )                          ⁢                              s            2            *                    ⁡                      (                          t              -              τ                        )                          ⁢                  ⅇ                                                                      -                  ⅈδ                                *                t                            -              τ                        )                          ⁢                  ⅆ          t                      ,where s1 and s2 each represent a signal reading, each of which may arise from a single emitted signal or be a composite of several component signals possibly originating from different signal emitters. The signal readings s1 and s2 may be radio frequency (“RF”) or downconverted intermediate frequency (“IF”). In the above equation, the symbol τ is a time parameter, δ is a frequency parameter, the star symbol (“*”) represents the complex conjugate, and T represents the time interval over which the measurements are taken. The symbols τ and δ are used in the above equation to represent time and frequency shift, respectively, between component signals of s1 and s2 that originate from a common emitter. The parameter τ in the above equation is related to time difference on arrival (“TDOA”) and to receiver-dependant delays. The parameter δ in the above equation is related to frequency difference on arrival (“FDOA”) and to downconversion shifts.
The representation of the CAF given above is for illustrative purposes and is not meant to be limiting. The CAF may take other forms, representations, or variants. By way of non-limiting example, one such form is a CAF that employs an additional term (e.g., β) for frequency-dependent Doppler shift. Such a form is particularly suited for broadband signals. Nevertheless, because such forms, representations, and variants are used to derive essentially the same information, the article “the” is used when referring to “CAF.” That is, any function that derives essentially the same information from essentially the same inputs is referred to herein as “the CAF.”
The two dimensions τ and δ in the above equation define a plane, which is referred to as “the CAF plane.” Other representations of the CAF plane that do not use these specific symbols are also possible. The values of the CAF for specific values of τ and δ defines a surface over the plane, and peaks on this surface represent a signal source. By scanning the CAF plane, values of τ and δ for one or more signal emitters may be determined. The actual locations of the signal source(s) may be derived from this information. Thus, the CAF is used in RADAR processing and geolocation techniques. Using the above terminology, in RADAR a signal is transmitted, s1 is received and s2 is a copy of the transmitted signal. The received signal s1 includes signal components reflected from different objects, each of which will arrive at a different delay and different Doppler (frequency shift), which information is used to determine the range and (at least a component of the) speed of each object relative to the RADAR transmitter.
The CAF is typically very computationally intensive to calculate, especially for broadband signals where a scale factor instead of, or in addition to, a frequency-shift term is used. Accordingly, standard analog or digital systems are relatively slow and expensive except for the narrowband case. Acousto-optical techniques have been proposed, but suffer from limited dynamic range and a very small TDOA search range. These and other drawbacks exist with current systems.
Two photons quantum mechanically entangled together are referred to as an entangled-photon pair (also, “biphotons”). Traditionally, the two photons comprising an entangled-photon pair are called “signal” and “idler” photons. The designations “signal” and “idler” are arbitrary and may be used interchangeably. The photons in an entangled photon pair have a connection between their respective properties. Measuring properties of one photon of an entangled-photon pair determines properties of the other photon, even if the two photons are separated by a distance. As understood by those of ordinary skill in the art and by way of non-limiting example, the quantum mechanical state of an entangled-photon pair cannot be factored into a tensor product of two individual quantum states.